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arXiv · 2208.11837

Hausdorff Dimension of Closure of Cycles in d-Maps on the Circle

Nicholas Payne·Mrudul Thatte
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Abstract

We study the dynamics of the map $x$ to $dx$ (mod 1) on the unit circle. We characterize the invariant finite subsets of this map which are called cycles and are graded by their degrees. By looking at the combinatorial properties of the base-d expansion of the elements in the cycles, we prove a conjecture of Curt McMullen that the Hausdorff dimension of the closure of degree-m cycles is equal to log $m$ / log $d$.

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Nicholas Payne, Mrudul Thatte. 2022-08-25. Hausdorff Dimension of Closure of Cycles in d-Maps on the Circle. https://arxiv.org/abs/2208.11837

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